Question

Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is : 

(i) intersecting lines 

(ii) parallel lines 

(iii) coincident lines.

Answer 1

Linear equation is 2x + 3y – 8 = 0. 

(i) Linear equations for intersecting lines: 

2x + 3y – 8 = 0 

3x + 2y – 7 = 0 

a₁ = 2, b₁ = 3, c₁ = -8 

a₂= 3, b₂ = 2, c₂ = -7

\(\frac{a_1}{a_2} = \frac{2}{3}\), \(\frac{b_1}{b_2} = \frac{3}{2}\)

Here, when \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)

Geometrical representation is intersecting lines. 

(ii) For Parallel lines : 

2x + 3y – 8 = 0 

2x + 3y – 12 = 0 

a₁ = 2, b₁ = 3, c₁= -8 

a₁ = 2, b₁ = 3, c₁ = -12

\(\frac{c_1}{c_2} = \frac{-8}{-12} = \frac{8}{12} = \frac{2}{3}\)

Here, hence 

Graphical representation is parallel lines. 

(iii) For Intersecting lines : 

2x + 3y – 8 = 0 

4x + 6y – 16 = 0 

a₁ = 2, b₁ = 3, c₁ = -8 

a₁ = 4, b₁ = 6, c₁ = -16

Here, 

∴ Lines are intersecting.